# Is a negative number rational

### What are natural and integers?

### Natural numbers $$ NN $$

The number range of the **natural numbers $$ NN $$** forms that **counting** as a natural process.

The

**smallest natural number**is the $$ 0 $$.The set of natural numbers contains all successors of the $$ 0 $$ up to infinity:

**$$ NN = {0,1,2,3,4, ..., n, n + 1, ...} $$**.

### How can you calculate with natural numbers?

You are allowed without restriction **add** and **multiply**.

- It is said that $$ NN $$ is related to addition and multiplication
**completed**. - All other arithmetic operations cannot be carried out without restrictions.

### Whole numbers $$ ZZ $$

If you expand the number range of the natural numbers with the **negative numbers**, do you have the whole numbers:

- In the set of negative numbers are
**all positive and negative numbers**without comma:**$$ ZZ = {…, -3, -2, -1,0,1,2,3,…} $$** - Now you can also without restrictions
**subtract**.

**Successor principle:** Is **$$ n $$** is any natural number, then **$$ n + 1 $$** her successor.

**Example:** The number $$ n = 73 $$ has the successor $$ n + 1 = 74 $$

**Seclusion:** The result of the calculation is the same amount, here $$ NN $$.

**Example:**

- If you add two natural numbers, the sum is also a natural number. $$ 4 + 3 = 7 $$
- If you calculate $$ 4: 3 $$, the result is not a natural number, but a fraction $$4/3$$.

### What are Fractional and Rational Numbers?

### Broken Numbers $$ QQ $$_{$$+$$}

Do you want unlimited **to divide**, you need the fractions.

- $$ QQ $$
_{$$+$$}contains**all positive fractions** - $$ QQ $$
_{$$+$$}$$ = $$$$ {$$$$ a / b | $$ $$ a, b $$ is a natural number and $$ b! = 0} $$

### Rational Numbers $$ QQ $$

Do you take the **negative fractions** in addition, you have the rational numbers.

- $$ QQ = {$$$$ a / b | $$ $$ a $$ is an integer $$, b $$ is a natural number and $$ b! = 0} $$
- In $$ QQ $$ you can
**all basic arithmetic**run without restriction. - $$ QQ $$ contains all positive and negative fractions, as well as all
**terminating**Decimal fractions (e.g. $$ - 3.75 $$) and**periodic**Decimal fractions (e.g. $$ 0.66666 ... $$).

##### You write a fraction generally $$ a / b $$.

##### The quotient of two natural numbers is positive.

##### Division by zero is not permitted in any number range, therefore $$ b! = 0 $$.

##### $$ a $$ can be negative, so the quotient can also be negative.

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### Number ranges in one another

The number ranges lie one inside the other.

For example, the $$ 2 $$ is a natural number, an integer, and a rational number.

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